Identifiability of large-scale non-linear dynamic network models applied to the ADM1-case study.

In this work, both the structural and practical identifiability of the Anaerobic Digestion Model no. 1 (ADM1) is investigated, which serves as a relevant case study of large non-linear dynamic network models. The structural identifiability is investigated using the probabilistic algorithm, adapted to deal with the specifics of the case study (i.e., a large-scale non-linear dynamic system of differential and algebraic equations). The practical identifiability is analyzed using a Monte Carlo parameter estimation procedure for a 'non-informative' and 'informative' experiment, which are heuristically designed. The model structure of ADM1 has been modified by replacing parameters by parameter combinations, to provide a generally locally structurally identifiable version of ADM1. This means that in an idealized theoretical situation, the parameters can be estimated accurately. Furthermore, the generally positive structural identifiability results can be explained from the large number of interconnections between the states in the network structure. This interconnectivity, however, is also observed in the parameter estimates, making uncorrelated parameter estimations in practice difficult.

Dynamic optimization of biological networks under parametric uncertainty.

BACKGROUND: Micro-organisms play an important role in various industrial sectors (including biochemical, food and pharmaceutical industries). A profound insight in the biochemical reactions inside micro-organisms enables an improved biochemical process control. Biological networks are an important tool in systems biology for incorporating microscopic level knowledge. Biochemical processes are typically dynamic and the cells have often more than one objective which are typically conflicting, e.g., minimizing the energy consumption while maximizing the production of a specific metabolite. Therefore multi-objective optimization is needed to compute trade-offs between those conflicting objectives. In model-based optimization, one of the inherent problems is the presence of uncertainty. In biological processes, this uncertainty can be present due to, e.g., inherent biological variability. Not taking this uncertainty into account, possibly leads to the violation of constraints and erroneous estimates of the actual objective function(s). To account for the variance in model predictions and compute a prediction interval, this uncertainty should be taken into account during process optimization. This leads to a challenging optimization problem under uncertainty, which requires a robustified solution. RESULTS: Three techniques for uncertainty propagation: linearization, sigma points and polynomial chaos expansion, are compared for the dynamic optimization of biological networks under parametric uncertainty. These approaches are compared in two case studies: (i) a three-step linear pathway model in which the accumulation of intermediate metabolites has to be minimized and (ii) a glycolysis inspired network model in which a multi-objective optimization problem is considered, being the minimization of the enzymatic cost and the minimization of the end time before reaching a minimum extracellular metabolite concentration. A Monte Carlo simulation procedure has been applied for the assessment of the constraint violations. For the multi-objective case study one Pareto point has been considered for the assessment of the constraint violations. However, this analysis can be performed for any Pareto point. CONCLUSIONS: The different uncertainty propagation strategies each offer a robustified solution under parametric uncertainty. When making the trade-off between computation time and the robustness of the obtained profiles, the sigma points and polynomial chaos expansion strategies score better in reducing the percentage of constraint violations. This has been investigated for a normal and a uniform parametric uncertainty distribution. The polynomial chaos expansion approach allows to directly take prior knowledge of the parametric uncertainty distribution into account.

Dynamic estimation of specific fluxes in metabolic networks using non-linear dynamic optimization.

BACKGROUND: Metabolic network models describing the biochemical reaction network and material fluxes inside microorganisms open interesting routes for the model-based optimization of bioprocesses. Dynamic metabolic flux analysis (dMFA) has lately been studied as an extension of regular metabolic flux analysis (MFA), rendering a dynamic view of the fluxes, also in non-stationary conditions. Recent dMFA implementations suffer from some drawbacks, though. More specifically, the fluxes are not estimated as specific fluxes, which are more biologically relevant. Also, the flux profiles are not smooth, and additional constraints like, e.g., irreversibility constraints on the fluxes, cannot be taken into account. Finally, in all previous methods, a basis for the null space of the stoichiometric matrix, i.e., which set of free fluxes is used, needs to be chosen. This choice is not trivial, and has a large influence on the resulting estimates. RESULTS: In this work, a new methodology based on a B-spline parameterization of the fluxes is presented. Because of the high degree of non-linearity due to this parameterization, an incremental knot insertion strategy has been devised, resulting in a sequence of non-linear dynamic optimization problems. These are solved using state-of-the-art dynamic optimization methods and tools, i.e., orthogonal collocation, an interior-point optimizer and automatic differentiation. Also, a procedure to choose an optimal basis for the null space of the stoichiometric matrix is described, discarding the need to make a choice beforehand. The proposed methodology is validated on two simulated case studies: (i) a small-scale network with 7 fluxes, to illustrate the operation of the algorithm, and (ii) a medium-scale network with 68 fluxes, to show the algorithm's capabilities for a realistic network. The results show an accurate correspondence to the reference fluxes used to simulate the measurements, both in a theoretically ideal setting with no experimental noise, and in a realistic noise setting. CONCLUSIONS: Because, apart from a metabolic reaction network and the measurements, no extra input needs to be given, the resulting algorithm is a systematic, integrated and accurate methodology for dynamic metabolic flux analysis that can be run online in real-time if necessary.